Inverse modeling of tight gas reservoirs [Elektronische Ressource] / von: George Mtchedlishvili

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Informations

• Publié par: technische_universitat_bergakademie_freiberg
• Publié le: 01 janvier 2007
• Nombre de lectures: 23
• Langue: English
• Poids de l'ouvrage: 2 Mo

Extrait

Inverse Modeling of Tight Gas Reservoirs



Der Fakultät für Geowissenschaften, Geotechnik und Bergbau
der Technischen Universität Bergakademie Freiberg
eingereichte
Dissertation
Zur Erlangung des akademischen Grades
Doktor-Ingenieur
Dr.-Ing.
vorgelegt




von: Dipl.-Ing. Mtchedlishvili George
geboren am 01.10.1977 in Tbilisi, Georgien
Freiberg, den: 25.01.2007 Abstract
The present thesis focuses on the following issues: (i) inverse modeling techniques for characterization
of tight-gas reservoirs, (ii) the numerical investigations of advanced well stimulation techniques, such
as hydraulic fracturing as well as underbalanced drilling, and (iii) the statistical analyses of results for
identification of the optimal level of parameterization for calibrated model as quality and quantity of
the measured data justifies.
In terms of a considerable increase the quality of characterization of tight-gas reservoirs, the aim of
this work was (i) an accurate representation of technological aspects and specific conditions in a reser-
voir simulation model, induced after the hydraulic fracturing or as a result of the underbalanced drill-
ing procedure and (ii) performing the history match on a basis of real field data to calibrate the gener-
ated model by identifying the main model parameters and to investigate the different physical
mechanisms, e.g. multiphase flow phenomena, affecting the well production performance.
Due to the complexity of hydrocarbon reservoirs and the simplified nature of the numerical model, the
study of the inverse problems in the stochastic framework provides capabilities using diagnostic statis-
tics to quantify a quality of calibration and the inferential statistics that quantify reliability of parame-
ter estimates. As shown in the present thesis the statistical criteria for model selection may help the
modelers to determine an appropriate level of parameterization and one would like to have as good an
approximation of the structure of the system as the information permits.

2 Acknowledgment
First of all, I would like to thank my supervisor, Prof. Frieder Häfner, for his very kind encourage-
ment, support and guidance during this work.
I am very indebted to Dr. Aron Behr for very useful discussions and for the positive influence on my
professional career.
I wish to express my sincere thanks and gratitude to Dr. Hans-Dieter Voigt and Dr. Torsten Friedel,
who gave useful contributions at various times during the development of this thesis.
I am grateful to the staff of Institute of Drilling Engineering and Fluid Mining, TU Bergakademie
Freiberg.
Finally, my warmest thanks go to my family for their effort, moral support and love.

3 Content
ABSTRACT 2
ACKNOWLEDGMENT 3
CONTENT 4
NOMENCLATURA 7
LIST OF FIGURES 11
LIST OF TABLES 13
1 STATEMENT OF THE PROBLEM 14
2 AN INTRODUCTION TO INVERSE PROBLEMS 17
2.1 Reservoir Modeling 17
2.2 The Inverse Modeling 19
2.2.1 The Statistical Methods for Model Calibration 20
2.2.1.1 Statistical Criteria of Parameter Estimation with Homogeneous Prior Distribution 23
2.2.1.2 Statistical Criteria of Parameter Estimation with Gaussian Prior Distribution 24
2.3 Numerical Optimization Algorithms 25
2.3.1 Nonlinear Last Squares – Gradient Based Methods 27
2.3.2 Global-Optimization Techniques – Direct Search Methods 29
2.3.2.1 Neighbourhood Algorithm 29
2.3.3 Analyses of Parameter Estimation Uncertainties 33
2.3.4 Optimality and Model Identification Criteria 35
2.3.4.1 Kullback-Leibler Information 35
2.3.4.2 Alternative Model Selection Criteria 38
2.3.5 Application to the Synthetic Field Example 39
2.3.5.1 Reservoir Description and Parameterization 39
2.3.5.2 Simulation Results 40
4 3 SIMULATION OF THE PRODUCTION BEHAVIOR OF HYDRAULICALLY
FRACTURED WELLS IN TIGHT GAS RESERVOIRS 46
3.1 Introduction to Fractured Well Simulation 47
3.2 Automatic Generaton of the Simulation Model for Fractured Wells 50
3.2.1 Local Grid Refinement 51
3.2.2 Integration of the Fracture Parameters 53
3.2.3 Estimation of the Fluid Distribution in the Invaded Zone 55
3.2.3.1 Evaluation of Exposure Time 55
3.3 History-Matching of a Case Study of Hydraulically Fractured Well 60
3.3.1 Field Example 60
3.3.2 Simulation Model 61
3.3.3 History Match 64
3.3.3.1 Cleanup with hydraulic damage 64
3.3.3.2 Combined effect of hydraulic and mechanical damage 67
3.3.4 Discussion of history match results 68
3.3.5 Effect of cleanup on postfracture performance 69
3.4 Identification of the Leakoff Coefficient by History Matching 71
3.4.1 Hypothetical Study 72
3.4.2 Case Study 76
3.5 Automatic Methods of Optimizing of Fractured Wells 79
4 SIMULATION OF INFLOW WHILEST UNDERBALANCED DRILLING (UBD)
WITH AUTOMATIC IDENTIFICATION OF FORMATION PARAMETERS AND
ASSESSMENT OF UNCERTAINTY 84
4.1 Literature review 85
4.2 Simulation of Reservoir Flow during UBD 86
4.3 Parameter Estimation during UBD 87
4.4 Identification Procedure 88
4.5 Uncertainty Analyses 90
4.6 Example 91
4.7 Outlook: Optimization Approach for UBD 93
5 5 SUMMARY 94
BIBLIOGRAPHY 97
6 Nomenclatura
Symbols
Symbol Meaning Unit
A area,m²
sensitivity matrix
a,b,c coefficients -
b set of subsidiary conditions
b fracture width m f
c proppant concentration kg/m²
1/2C leakoff coefficient m/sl
* 1/2 γ C specific leakoff coefficient m/s mDl
Cov prior parameter-covariance matrix 0
Cov covariance matrix of measured values z
d distance, m
step size
F fractional flow, dimensionless -
F fracture conductivity mD*m C
F dimensionless fracture conductivity -CD
g Gradientvector
H Hessianmatrix
J Jacobian matrix
I Unitmatrix
k permeability, m², Darcy
index -
L linear size, m
M number of gridblocks -
model -
N number of pumping periods -
N number of measured data m
N number of model parameters p
n coordinate normal to fracture plane m
n ,n number of Voronoi cells s r
p pressure, Pa
set of physical parameters
R number
7 r index
right-hand-side term m³
q mass source/sink kg/(m³*s)
set of control variables -
S saturation -
s estimated error variance 0
t time sec
u set of state varaibles
V volume
x coordinate, m
set of spatial and time variables
x fracture half length m f

Greek Symbols
Symbol Meaning Unit
power factor, - α
fracture expansion factor
porosity - φ
difference, increment - ∆, δ
error ε
power factor - γ
net-to-gross thickness ratio - η
dynamic viscosity Pa*s µ
viscosity ratio - µ0
displacement direction, - λ
regularization factor,
material parameter
model parameter vector Θ
density kg/m³ ρ
effective stress, MPaσ
standard deviation
velocity m/s ν
domain Ω
weight factor ω
self-similar variable ξ
8 Indices
Index Meaning
0 initial
a absolute
ad admissible
cap capillary
calclculated
const constant
f fracture,
fictive
g gas
i index
phase index,
index of time period
j index
index of time period
k
meas measured
obs observed
p proppant
pr primary
prededictive
r relative
ref reference
s slurry
t total
w water

9 Functions and Operators
Function Meaning
AIC Akaike Information Criterion
BIC Bayesian Information Criterion
d Kashyap Index m
J Leverett J-Function
H statistical entropy
I information content
L partial differentialoperator,
likelihood
p probability density function
p prior distribution 0
p posterior distribution *
divergence operator ∇.


Abbreviations
Abbreviation Meaning
BHP Bottom Hole Pressure
CSS Composite Scaled Sensitivities
DF Discount rate
FOPT Field Oil Production Total
GOR Gas-OilRatio
GPR Gas Production Rate
LGR Local Grid Refinement
NPV Net PresentValue
OF ObjectiveFunction
THP Tubing Head Pressure
TOL Tolerance Criteria
UBD UnderbalancedDrilling
WGR Water Gas Ratio

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